AP Calculus BC : Chain Rule and Implicit Differentiation

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

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Example Question #1 : Chain Rule And Implicit Differentiation

Find dy/dx by implicit differentiation:

Possible Answers:

Correct answer:

Explanation:

To find dy/dx we must take the derivative of the given function implicitly. Notice the term  will require the use of the Product Rule, because it is a composition of two separate functions multiplied by each other. Every other term in the given function can be derived in a straight-forward manner, but this term tends to mess with many students. Remember to use the Product Rule:

Product Rule: 

 

Now if we take the derivative of each component of the given problem statement:

Notice that anytime we take the derivative of a term with involved we place a "dx/dx" next to it, but this is equal to "1".

So this now becomes:

Now if we place all the terms with a "dy/dx" onto one side and factor out we can solved for it:

This is one of the answer choices.

 

Example Question #2 : Chain Rule And Implicit Differentiation

Find dx/dy by implicit differentiation:

Possible Answers:

Correct answer:

Explanation:

To find dx/dy we must take the derivative of the given function implicitly. Notice the term  will require the use of the Product Rule, because it is a composition of two separate functions multiplied by each other. Every other term in the given function can be derived in a straight-forward manner, but this term tends to mess with many students. Remember to use the Product Rule:

Product Rule: 

 

Now if we take the derivative of each component of the given problem statement:

Notice that anytime we take the derivative of a term with y involved we place a "dy/dy" next to it, but this is equal to "1".

So this now becomes:

Now if we place all the terms with a "dx/dy" onto one side and factor out we can solved for it:

This is one of the answer choices.

Example Question #7 : Derivative At A Point

Use implicit differentiation to find the slope of the tangent line to  at the point .

Possible Answers:

Correct answer:

Explanation:

We must take the derivative  because that will give us the slope. On the left side we'll get 

, and on the right side we'll get .

We include the  on the left side because  is a function of , so its derivative is unknown (hence we are trying to solve for it!).

Now we can factor out a  on the left side to get 

 and divide by  in order to solve for .

Doing this gives you

 .

We want to find the slope at , so we can sub in  for  and 

.

Example Question #72 : Derivative Review

Evaluate .

Possible Answers:

Correct answer:

Explanation:

To find , substitute  and use the chain rule:

 

Plug in 3:

Example Question #73 : Derivative Review

Evaluate .

Possible Answers:

Undefined

Correct answer:

Explanation:

To find , substitute  and use the chain rule: 

So

and

Example Question #74 : Derivative Review

Evaluate .

Possible Answers:

Undefined

Correct answer:

Explanation:

To find , substitute  and use the chain rule:

 

 

Example Question #75 : Derivative Review

Evaluate .

Possible Answers:

Correct answer:

Explanation:

To find , substitute  and use the chain rule: 

So 

and

Example Question #3 : Chain Rule And Implicit Differentiation

Evaluate .

Possible Answers:

Correct answer:

Explanation:

To find , substitute  and use the chain rule:

So 

and 

Example Question #8 : Chain Rule And Implicit Differentiation

Possible Answers:

Correct answer:

Explanation:

Consider this function a composition of two functions, f(g(x)). In this case, f(x) is ln(x) and g(x) is 3x - 7. The derivative of ln(x) is 1/x, and the derivative of 3x - 7 is 3. The derivative is then 

Example Question #3 : Chain Rule And Implicit Differentiation

Possible Answers:

Correct answer:

Explanation:

Consider this function a composition of two functions, f(g(x)). In this case,  and . According to the chain rule, . Here,  and , so the derivative is 

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